In-plane spin excitation anisotropy in the paramagnetic state of NaFeAs

We use unpolarized and polarized inelastic neutron scattering to study low-energy spin excitations in NaFeAs, which exhibits a tetragonal-to-orthorhombic lattice distortion at $T_s\approx 58$ K followed by a collinear antiferromagnetic (AF) order below $T_N\approx 45$ K. In the AF ordered state ($T<T_N$), spin waves are entirely $c$-axis polarized below $\sim$10 meV, exhibiting a gap of $\sim$4 meV at the AF zone center and disperse to $\sim$7 meV near the $c$-axis AF zone boundary. On warming to the paramagnetic state with orthorhombic lattice distortion ($T_N<T<T_s$), spin excitations become anisotropic within the FeAs plane. Upon further warming to the paramagnetic tetragonal state ($T>T_s$), spin excitations become more isotropic. Since similar magnetic anisotropy is also observed in the paramagnetic tetragonal phase of superconducting BaFe${}_{1.904}$Ni${}_{0.096}$As${}_2$, our results suggest that the spin excitation anisotropy in superconducting iron pnictides originates from similar anisotropy already present in their parent compounds.

Figure 1

(a) The orthorhombic chemical unit cell of NaFeAs (enclosed by gray lines, two orthorhombic chemical unit cells are stacked along the $c$ axis). The black arrows indicate directions of the ordered moments and orange spheres represent Fe atoms, Na and As atoms are not shown. (b) Points in reciprocal space probed in this work. The points represent magnetic zones centers ($L=0.5,1.5$) while the crosses represent magnetic zone boundaries ($L=0,1$) along the $(1,0,L)$ direction. The dashed green lines enclose the magnetic Brillouin zone. The relationship between the neutron polarization directions ($x,y,z$) and the scattering plane are shown in (c) for $\mathbf{Q}=(1,0,0.5)$ and in (e) for $(1,0,1.5)$. The angle between $x$ direction and the $H$ axis is denoted as ${\theta }_1$ for $(1,0,0.5)$ and ${\theta }_2$ for $(1,0,1.5)$. The magnetic components measured in each channel and their relationships to the magnetic components projected onto the crystal axes are shown in (d) for $\mathbf{Q}=(1,0,0.5)$ and in (f) for $(1,0,1.5)$. Only SF channels are measured in this work. ${\sigma }_x^{\mathrm{SF}}$ contains both $M_y$ and $M_z$ magnetic components, whereas only $M_y$ and $M_z$ contribute to ${\sigma }_z^{\mathrm{SF}}$ and ${\sigma }_y^{\mathrm{SF}}$, respectively. The relationships between the magnetic components along the neutron polarization directions ($M_x$, $M_y$, and $M_z$) and crystal axes ($M_a$, $M_b$, and $M_c$) are shown in the boxes in (d) and (f) for $\mathit{L}=0.5$ and $\mathit{L}=1.5$, respectively.

Figure 2

(a) Constant-$\mathbf{Q}$ scans at $(1,0,\mathit{L})$ at 4 K for $\mathit{L}=0,0.2,0.3$, and 0.5 with unpolarized neutrons after background has been subtracted. Solid lines are fits to the response of a damped harmonic oscillator convolved with instrumental resolution with low-energy spin waves modeled as $E\left(\mathbf{Q}\right)=\sqrt{\Delta {(1,0,\mathit{L})}^2+v_a^2{h}^2+v_b^2{k}^2}$, where $(h,k)=(H,K)-(1,0)$.[38] Data points with $E<11$ meV are used in the fits. (b) Dispersion of the low-energy spin waves along the $(1,0,\mathit{L})$ direction, points are fitted gap values $\Delta (1,0,\mathit{L})$, the purple solid line is the dispersion from linear spin wave theory using effective exchange couplings from Ref. 39.

Figure 3

Constant-$\mathbf{Q}$ scans in the AF ordered state ($T=2$ K) at $\mathbf{Q}=(1,0,L)$ with $\mathit{L}=0,0.5,1$, and 1.5 are shown in (b), (a), (d), and (c), respectively. All three spin-flip channels are measured, denoted as ${\sigma }_x^{\mathrm{SF}}$, ${\sigma }_y^{\mathrm{SF}}$, and ${\sigma }_z^{\mathrm{SF}}$ for the three different neutron polarization directions. From the measured cross sections at $(1,0,0.5)$ and $(1,0,1.5)$, the magnetic components at the magnetic zone center along the crystallographic axes $M_a$, $M_b$, and $M_c$ are determined and plotted in (e). Similarly, (f) shows the magnetic components along different crystallographic axes for magnetic zone boundaries at $\mathbf{Q}=(1,0,0)$ and $(1,0,1)$. The solid lines in (e) and (f) are guides to the eye.

Figure 4

Constant-$\mathbf{Q}$ scans at $T=60$ K for $(1,0,0.5)$ and $(1,0,1.5)$ are shown in (a) and (b), respectively. Temperature dependence of the three spin-flip cross sections for the AF zone centers $(1,0,0.5)$ and $(1,0,1.5)$ are shown in (c) and (d). (e) The difference of the magnetic components $M_y$ and $M_z$ for $\mathit{L}=0.5$ determined from ${\sigma }_z^{\mathrm{SF}}$ and ${\sigma }_y^{\mathrm{SF}}$ in (c). The temperature dependence of $M_a$, $M_b$, and $M_c$ at the magnetic zone center are determined and plotted in (f). In cases where only cross sections for $\mathit{L}=0.5$ are measured, only $M_b$ can be determined, when ${\sigma }_x^{\mathrm{SF}}$ and ${\sigma }_y^{\mathrm{SF}}$ for $\mathit{L}=1.5$ are in addition measured, all three components along crystal axes can be determined, when all three channels are measured for both $\mathit{L}=0.5$ and $\mathit{L}=1.5$, the overdetermination is used to improve estimates of $M_a$, $M_b$, and $M_c$. The point at $T\approx 60$ K is obtained by combining raw data from temperatures in the range indicated by horizontal bars and the constant-$\mathbf{Q}$ scans in (a) and (b) from $E=4$–8 meV, the combined ${\sigma }_x^{\mathrm{SF}}$, ${\sigma }_y^{\mathrm{SF}}$, and ${\sigma }_z^{\mathrm{SF}}$ for $\mathit{L}=0.5$ and $\mathit{L}=1.5$ are shown in the inset. The solid vertical gray line through (c), (d), (e), and (f) marks the magnetic transition temperature $T_N$, whereas the dashed gray line marks the structural transition temperature $T_s$. The solid lines in (e) and (f) are guides to the eye.